Q: What is the prime factorization of the number 12,172,160?

 A:
  • The prime factors are: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 5 x 7 x 11 x 13 x 19
    • or also written as { 2, 2, 2, 2, 2, 2, 2, 5, 7, 11, 13, 19 }
  • Written in exponential form: 27 x 51 x 71 x 111 x 131 x 191

Why is the prime factorization of 12,172,160 written as 27 x 51 x 71 x 111 x 131 x 191?

What is prime factorization?

Prime factorization or prime factor decomposition is the process of finding which prime numbers can be multiplied together to make the original number.

Finding the prime factors of 12,172,160

To find the prime factors, you start by dividing the number by the first prime number, which is 2. If there is not a remainder, meaning you can divide evenly, then 2 is a factor of the number. Continue dividing by 2 until you cannot divide evenly anymore. Write down how many 2's you were able to divide by evenly. Now try dividing by the next prime factor, which is 3. The goal is to get to a quotient of 1.

If it doesn't make sense yet, let's try it...

Here are the first several prime factors: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29...

Let's start by dividing 12,172,160 by 2

12,172,160 ÷ 2 = 6,086,080 - No remainder! 2 is one of the factors!
6,086,080 ÷ 2 = 3,043,040 - No remainder! 2 is one of the factors!
3,043,040 ÷ 2 = 1,521,520 - No remainder! 2 is one of the factors!
1,521,520 ÷ 2 = 760,760 - No remainder! 2 is one of the factors!
760,760 ÷ 2 = 380,380 - No remainder! 2 is one of the factors!
380,380 ÷ 2 = 190,190 - No remainder! 2 is one of the factors!
190,190 ÷ 2 = 95,095 - No remainder! 2 is one of the factors!
95,095 ÷ 2 = 47,547.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
95,095 ÷ 3 = 31,698.3333 - This has a remainder. 3 is not a factor.
95,095 ÷ 5 = 19,019 - No remainder! 5 is one of the factors!
19,019 ÷ 5 = 3,803.8 - There is a remainder. We can't divide by 5 evenly anymore. Let's try the next prime number
19,019 ÷ 7 = 2,717 - No remainder! 7 is one of the factors!
2,717 ÷ 7 = 388.1429 - There is a remainder. We can't divide by 7 evenly anymore. Let's try the next prime number
2,717 ÷ 11 = 247 - No remainder! 11 is one of the factors!
247 ÷ 11 = 22.4545 - There is a remainder. We can't divide by 11 evenly anymore. Let's try the next prime number
247 ÷ 13 = 19 - No remainder! 13 is one of the factors!
19 ÷ 13 = 1.4615 - There is a remainder. We can't divide by 13 evenly anymore. Let's try the next prime number
19 ÷ 17 = 1.1176 - This has a remainder. 17 is not a factor.
19 ÷ 19 = 1 - No remainder! 19 is one of the factors!

The orange divisor(s) above are the prime factors of the number 12,172,160. If we put all of it together we have the factors 2 x 2 x 2 x 2 x 2 x 2 x 2 x 5 x 7 x 11 x 13 x 19 = 12,172,160. It can also be written in exponential form as 27 x 51 x 71 x 111 x 131 x 191.

Factor Tree

Another way to do prime factorization is to use a factor tree. Below is a factor tree for the number 12,172,160.

12,172,160
Factor Arrows
26,086,080
Factor Arrows
23,043,040
Factor Arrows
21,521,520
Factor Arrows
2760,760
Factor Arrows
2380,380
Factor Arrows
2190,190
Factor Arrows
295,095
Factor Arrows
519,019
Factor Arrows
72,717
Factor Arrows
11247
Factor Arrows
1319

More Prime Factorization Examples

12,172,15812,172,15912,172,16112,172,162
21 x 32 x 671 x 10,09313,2211 x 3,779131 x 7271 x 5,581121 x 411 x 791 x 1,8791

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